Non-commutators in finite groups

DefinitionKourovka Problem 17.76 (finite groups with exactly one non-commutator)#✓fully proved

This is the statement that there is a finite group \(G\) of order greater than \(2\) in which exactly one element is not a commutator, that is, not of the form \(x^{-1}y^{-1}xy\) with \(x, y \in G\).

def exactlyOneNoncommutator : Prop :=
  ∃ (G : Type) (_ : Group G) (_ : Finite G), 2 < Nat.card G ∧
    {g : G | ¬ ∃ x y : G, g = x⁻¹ * y⁻¹ * x * y}.ncard = 1

Used by 1

Used byA finite group with exactly one non-commutator

TheoremA finite group with exactly one non-commutator#●ready to prove

There is a finite group \(G\) of order greater than \(2\) in which exactly one element is not a commutator, that is, not of the form \(x^{-1}y^{-1}xy\) with \(x, y \in G\).

Resolved true: Skresanov constructs an infinite series of finite groups with exactly one element that is not a commutator, the smallest of order 16609443840, and Hatem and Siniora find two such groups of order 368640, the smallest possible order. The Kourovka Notebook (21st issue) answers 17.76 "Yes".

public theorem_wanted exactlyOneNoncommutator_holds : exactlyOneNoncommutator

Sources

• E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Problem 17.76 (D. MacHale); resolution: S. V. Skresanov, Siberian Electron. Math. Rep. 23 (2026), 603-607 (preprint On finite groups with exactly one noncommutator, arXiv:2509.17587); O. Hatem and D. Siniora, Int. J. Group Theory 15 (2026), 161-167 (preprint A group with exactly one noncommutator, arXiv:2511.00541), Skresanov, abstract (an infinite series of such groups, the smallest of order 16609443840); Hatem-Siniora, abstract (two non-isomorphic groups of order 368640, the smallest order); the Kourovka Notebook (21st issue) marks 17.76 solved (Yes, such groups do exist, and there are infinitely many examples), https://arxiv.org/abs/2509.17587.

Uses 1

Statement usesKourovka Problem 17.76 (finite groups with exactly one non-commutator)

1 fully proved1 ready to prove

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